Derived categories of NDG categories

We study $N$-differential graded ($NDG$) categories and their the derived categories. First, we introduce $N$-differential modules over an $NDG$ category $\mathcal{A}$. Then we show that the category $\mathsf{C}_{Ndg}(\mathcal{A})$ of $N$-differential $\mathcal{A}$-modules is a Frobenius category, and that its homotopy category $\mathsf{K}_{Ndg}(\mathcal{a})$ is a triangulated category. Second, we study the properties of the derived category $\mathsf{D}_{Ndg}(\mathcal{A})$ and give triangle equivalences of Morita type between derived categories of $NDG$ categories. Finally, we show $\mathsf{D}_{Ndg}(\mathcal{A})$ is triangle equivalent to the derived category of some ordinary DG category.